A010030 Irregular triangle read by rows: T(n,k) (n >= 1, 0 <= k <= [n/2]) = number of permutations of 1..n with [n/2]-k runs of consecutive pairs up and down (divided by 2).

1, 1, 0, 3, 0, 3, 8, 1, 25, 28, 7, 17, 155, 143, 45, 259, 1005, 933, 323, 131, 2770, 7488, 7150, 2621, 3177, 27978, 64164, 62310, 23811, 1281, 51433, 294602, 619986, 607445, 239653

Offset

1,4

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.

Links

Table of n, a(n) for n=1..35.

Formula

G.f. for number of permutations of 1..n by number of runs of consecutive pairs up and down is Sum(n!*(((1-y)*(2*x^2-x^3)-x)/((1-y)*x^2-1))^n,n = 0 .. infinity), cf. A010029. - Vladeta Jovovic, Nov 23 2007

Example

Triangle begins:
1,
1, 0,
3, 0,
3, 8, 1,
25, 28, 7,
17, 155, 143, 45,
259, 1005, 933, 323,
131, 2770, 7488, 7150, 2621,
3177, 27978, 64164, 62310, 23811,
1281, 51433, 294602, 619986, 607445, 239653,
...

Crossrefs

Cf. A002464, A001266, A000239, A000544, A001282.

Sequence in context: A021771 A154853 A139214 * A197270 A117940 A099093

Adjacent sequences: A010027 A010028 A010029 * A010031 A010032 A010033

Keyword

tabf,nonn

Author

N. J. A. Sloane.

Extensions

More terms from Vladeta Jovovic, Nov 23 2007

Entry revised by N. J. A. Sloane, Apr 14 2014

Status

approved